Transcript Characterizing distribution rules for cost sharing games

Characterizing distribution rules for cost sharing games
Raga Gopalakrishnan
Caltech
Joint work with Jason R. Marden & Adam Wierman
Cost sharing games:
Self-interested agents make decisions, and
share the incurred cost among themselves.
Key Question: How should the cost be shared?
Lots of examples:
Network formation games
Facility location games
Profit sharing games
Cost sharing games:
Self-interested agents make decisions, and
share the incurred cost among themselves.
Key Question: How should the cost be shared?
Lots of examples:
Network formation games
Facility location games
Profit sharing games
D1
S1
S2
D2
Cost sharing games:
Self-interested agents make decisions, and
share the incurred cost among themselves.
Key Question: How should the cost be shared?
Lots of examples:
Network formation games
Facility location games
Profit sharing games
Cost sharing games:
Self-interested agents make decisions, and
share the incurred cost among themselves.
Key Question: How should the cost be shared?
Lots of examples:
Network formation games
Facility location games
Profit sharing games
Cost sharing games:
Self-interested agents make decisions, and
share the incurred cost among themselves.
Key Question: How should the cost be shared?
Lots of examples:
Network formation games
[Jackson 2003][Anshelevich et al. 2004]
Facility location games
[Goemans et al. 2000] [Chekuri et al. 2006]
Profit sharing games
Huge literature
in Economics
Growing
literature in CS
[Kalai et al. 1982] [Ju et al. 2003]
New application: Designing for distributed control
[Gopalakrishnan et al. 2011][Ozdaglar et al. 2009][Alpcan et al. 2009]
Cost sharing games (more formally):
set of resources
𝐺 = ( 𝑁, 𝑅, 𝒜𝑖
set of agents/players
utility function
of agent 𝑖
𝒰𝑖 (𝑎1 , 𝑎2 , … , 𝑎𝑛 )
welfare function
𝒲(𝑎1 , 𝑎2 , … , 𝑎𝑛 )
𝑖∈𝑁 , 𝒲,
𝒰𝑖
𝑖∈𝑁
action set of agent 𝑖
𝒜𝑖 ⊆ 2𝑅
)
D1
S1
Example:
S2
D2
Cost sharing games (more formally):
set of resources
𝐺 = ( 𝑁, 𝑅, 𝒜𝑖
set of agents/players
utility function
of agent 𝑖
𝒰𝑖 (𝑎1 , 𝑎2 , … , 𝑎𝑛 )
welfare function
𝒲(𝑎1 , 𝑎2 , … , 𝑎𝑛 )
𝑖∈𝑁 , 𝒲,
𝒰𝑖
𝑖∈𝑁
)
action set of agent 𝑖
𝒜𝑖 ⊆ 2𝑅
Assumption: 𝒲 is separable across resources
𝒲 𝑎 =
𝒲𝑟 ({𝑎}𝑟 )
𝑟∈R
set of agents choosing
resource 𝑟 in allocation 𝑎
Cost sharing games (more formally):
set of resources
welfare function
at resource 𝑟
𝒲𝑟 : 2𝑁 → ℝ
𝐺 = ( 𝑁, 𝑅, 𝒜𝑖
set of agents/players
𝑖∈𝑁 ,
𝒲𝑟
utility function
of agent 𝑖
𝒰𝑖 (𝑎1 , 𝑎2 , … , 𝑎𝑛 )
𝑟∈𝑅 ,
𝒰𝑖
𝑖∈𝑁
action set of agent 𝑖
𝒜𝑖 ⊆ 2𝑅
Assumption: 𝒲𝑟 is scalable
𝒲𝑟 ⋅ = 𝑣𝑟 𝑊 ⋅
𝑣𝑟 ∈ ℝ++
common base
welfare function
)
Cost sharing games (more formally):
set of resources
resource-specific
coefficients
𝐺 = ( 𝑁, 𝑅, 𝒜𝑖
set of agents/players
𝑖∈𝑁 ,
𝑣𝑟
utility function
of agent 𝑖
𝒰𝑖 (𝑎1 , 𝑎2 , … , 𝑎𝑛 )
𝑟∈𝑅 , 𝑊,
action set of agent 𝑖
𝒜𝑖 ⊆ 2𝑅
𝒰𝑖
𝑖∈𝑁
welfare
function
)
Cost sharing games (more formally):
resource-specific
coefficients
set of resources
𝐺 = ( 𝑁, 𝑅, 𝒜𝑖
set of agents/players
𝑖∈𝑁 ,
𝑣𝑟
utility function
of agent 𝑖
𝒰𝑖 (𝑎1 , 𝑎2 , … , 𝑎𝑛 )
𝑟∈𝑅 , 𝑊,
action set of agent 𝑖
𝒜𝑖 ⊆ 2𝑅
𝒰𝑖
𝑖∈𝑁
)
welfare
function
Assumption: Utility functions are also separable/scalable
𝒰𝑖 𝑎 =
𝑣𝑟 𝑓(𝑖, 𝑎 𝑟 )
𝑟∈𝑎𝑖
common base distribution rule
(portion of welfare at 𝑟 to agent 𝑖)
Cost sharing games (more formally):
set of resources
resource-specific
coefficients
𝐺 = ( 𝑁, 𝑅, 𝒜𝑖
set of agents/players
𝑖∈𝑁 ,
𝑣𝑟
distribution
rule
𝑟∈𝑅 , 𝑊, 𝑓
action set of agent 𝑖
𝒜𝑖 ⊆ 2𝑅
Goal: Design the distribution rule 𝑓
)
welfare
function
Requirements on the distribution rule
The distribution rule should be:
(i) Budget-balanced
(ii) “Stable” and/or “Fair”
(iii) “Efficient”
Requirements on the distribution rule
The distribution rule should be:
(i) Budget-balanced
(ii) “Stable” and/or “Fair”
(iii) “Efficient”
𝑓 𝑖, 𝑆 = 𝑊(𝑆)
𝑖∈𝑆
Requirements on the distribution rule
The distribution rule should be:
(i) Budget-balanced
(ii) “Stable” and/or “Fair”
(iii) “Efficient”
Lots of work on characterizing
“stability” and “fairness”
Core
[Gillies 1959]
[Devanur et al. 2003]
[Chander et al. 2006]
Nash
equilibrium
[von Neumann et al. 1944] [Nash 1951]
[Moulin 1992]
[Albers et al. 2006]
Requirements on the distribution rule
The distribution rule should be:
(i) Budget-balanced
(ii) “Stable” and/or “Fair”
(iii) “Efficient”
Lots of work on characterizing
“stability” and “fairness”
Core
[Gillies 1959]
[Devanur et al. 2003]
[Chander et al. 2006]
Nash
equilibrium
[von Neumann et al. 1944] [Nash 1951]
[Moulin 1992]
[Albers et al. 2006]
Requirements on the distribution rule
The distribution rule should be:
(i) Budget-balanced
(ii) “Stable” and/or “Fair”
(iii) “Efficient”
Has good Price of Anarchy and
Price of Stability properties
The Shapley value [Shapley 1953]
A player’s share of the welfare should depend on their
“average” marginal contribution
|𝑇| !
𝑓 𝑖, 𝑆 =
𝑇⊆𝑆\{𝑖}
𝑆 − 𝑇 −1 !
𝑊 𝑇∪ 𝑖
|𝑆| !
− 𝑊(𝑇)
Example: If players are homogeneous, 𝑓 𝑖, 𝑆 = 𝑊(𝑆)/|𝑆|
Note: There is also a weighted Shapley value
Players are assigned ‘weights’ 𝜔𝑖
Properties of the Shapley value
+ Guaranteed to be in the core for “balanced” games
[Shapley 1967]
+ Results in a potential game
[Ui 2000]
+ Guarantees the existence of a Nash equilibrium
- Often intractable to compute
approximations are
often tractable
[Conitzer et al. 2004]
- Not “efficient” in terms of social welfare
e.g. Price of Anarchy/Stability ≥ 2
[Marden et al. 2011]
[Castro et al. 2009]
Research question:
Are there distribution rules besides the (weighted) Shapley
value that always guarantee a Nash equilibrium?
If so: can designs be more efficient and/or more tractable?
If not: we can optimize over 𝜔𝑖 to determine the best design!
Research question:
Are there distribution rules besides the (weighted) Shapley
value that always guarantee a Nash equilibrium?
Our (surprising) answer:
NO, for any submodular welfare function.
“decreasing marginal returns”
natural way to model
many real-world problems
The inspiration for our work
Theorem (Chen, Roughgarden, Valiant):
There exists a welfare function 𝑊, for which no
distribution rules other than the weighted Shapley
value guarantee a Nash equilibrium in all games.
[Chen et al. 2010]
Our result
A game is specified by (𝑁, 𝑅, 𝒜𝑖
Theorem:
For any submodular welfare function 𝑊 , no
distribution rules other than the weighted Shapley
value guarantee a Nash equilibrium in all games.
𝑖∈𝑁 ,
𝜈𝑟
𝑟∈𝑅 , 𝑊, 𝑓)
The inspiration for our work
Theorem (Chen, Roughgarden, Valiant):
Given 𝑁, 𝑓, all games (𝑁, −, −, −, −, 𝑓) posses a
Nash equilibrium if and only if 𝑓 is a weighted
Shapley value.
[Chen et al. 2010]
Our result
Theorem:
Given 𝑁, 𝑓, and any submodular 𝑊, all games
(𝑁, −, −, −, 𝑊, 𝑓) posses a Nash equilibrium if and
only if 𝑓 is a weighted Shapley value.
Our result
Theorem:
For any submodular welfare function 𝑊 , no
distribution rules other than the weighted Shapley
value guarantee a Nash equilibrium in all games.
Consequences
 Can obtain the best distribution rule by optimizing the
player weights, 𝜔𝑖
 Can always work within a potential game
• Small, well-defined class of games
• Several learning algorithms for Nash equilibrium
 Fundamental limits on tractability and efficiency
Proof Sketch
First step: Represent 𝑊 using a linear basis
– Define a 𝑇-welfare function: 𝑊 𝑇
1
𝑇⊆𝑆
𝑆 =
0 otherwise
– Given any 𝑊, there exists a set 𝒯 ⊆ 2𝑁 , and a sequence
𝒬 = {𝑞𝑇 } 𝑇∈𝒯 of weights indexed by 𝒯, such that:
𝑞𝑇 𝑊 𝑇
𝑊≔
𝑇∈𝒯
“contributing
coalition”
“magnitude of
contribution”
Proof technique: Establish a series of necessary conditions on 𝑓
Proof Sketch (A single T-Welfare Function)
Proof technique: Establish a series necessary conditions on 𝑓
𝑇 is not formed in 𝑆
What is required
of 𝑓 𝑖, 𝑆 ?
𝑇 is formed in 𝑆
Don’t allocate welfare
to any player
Allocate welfare only
to players in 𝑇,
independent of others
⇓
⇓
𝑓 is completely specified by 𝑓(𝑖, 𝑇)
𝜔𝑖 =
𝑓(𝑖, 𝑇)
𝑖∈𝑇
𝑞𝑇
arbitrary 𝑖 ∉ 𝑇
𝑓 is a weighted Shapley value
Proof Sketch (General Welfare Functions)
Proof technique: Establish a series necessary conditions on 𝑓
What is required
of 𝑓 𝑖, 𝑆 ?
no coalition from
𝒯 is formed in 𝑆
Don’t allocate welfare
to any player
a coalition from
𝒯 is formed in 𝑆
Allocate welfare only
to players in these
formed coalitions,
independent of others
⇓
𝑓 𝑇 is the basis
weighted Shapley
value corresponding
to 𝑊 𝑇 , with weights 𝝎𝑇
𝑞𝑇 𝑓 𝑇
𝑓≔
𝑇∈𝒯
Key challenge: Each basis 𝑓 𝑇 might use different 𝝎𝑇 !
Proof Sketch (General Welfare Functions)
Proof technique: Establish a series necessary conditions on 𝑓
What is required
of 𝑓 𝑖, 𝑆 ?
no coalition from
𝒯 is formed in 𝑆
Don’t allocate welfare
to any player
a coalition from
𝒯 is formed in 𝑆
Allocate welfare only
to players in these
formed coalitions,
independent of others
⇓
Weights of common players
in any two coalitions must
be linearly dependent
⇓
𝑓 is a weighted
Shapley value
𝑊 is submodular
𝑞𝑇 𝑓 𝑇
𝑓≔
𝑇∈𝒯
Cost Sharing Games
Research question:
Are there distribution rules besides the (weighted) Shapley
value that always guarantee a Nash equilibrium?
Our answer:
NO, for any submodular welfare function.
what about for other welfare functions?
Understand what causes this fundamental
restriction – perhaps some structure of action sets?
Characterizing distribution rules for cost sharing games
Raga Gopalakrishnan
Caltech
Joint work with Jason R. Marden & Adam Wierman
References
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[von Neumann et al. 1944]
[Nash 1951]
[Shapley 1953]
[Gillies 1959]
[Shapley 1967]
[Kalai et al. 1982]
[Moulin 1992]
[Goemans et al. 2000]
[Ui 2000]
[Devanur et al. 2003]
[Jackson 2003]
[Ju et al. 2003]
[Anshelevich et al. 2004]
[Conitzer et al. 2004]
[Albers et al. 2006]
[Chander et al. 2006]
[Chekuri et al. 2006]
[Alpcan et al. 2009]
[Ozdaglar et al. 2009]
[Chen et al. 2010]
[Gopalakrishnan et al. 2011]
[Marden et al. 2011]